Transcript Analemma

Analemma
The Apparent Local Trajectory
of the Sun
Image: http://antwrp.gsfc.nasa.gov/apod/ap071002.html
Ana-what?

Take picture of the sun at same time every day
for a year

Superimpose pictures

Result forms “figure 8” on the sky.


Called analemma – the apparent trajectory of the
sun
Code will calculate this for any planet, but for
simplicity, we stick to the earth for now.
Analemma from Earth
Daylight
Daylight Removed
True Anomaly
Actual angle between
Earth and Sun.
Mean Anomaly
Angle between Earth
and Sun if Earth
followed circular orbit.
Mean anomaly and true
anomaly are identical at
perihelion and aphelion.
Eccentric Anomaly
Angle between direction
of periapsis and current
position projected onto
circular orbit.
How to Calculate True Anomaly
P = orbital period
2 t
M
P
t = time since perihelion
M = mean anomaly
T = true anomaly
E = eccentric anomaly
M  E  e sin E
e = eccentricity
 1 e
 E 
T  arctan  2
tan   
 2 
 1 e
Equation of Time
(due to eccentricity)

Equation of time due to eccentricity of
Earth’s orbit arises from difference
between mean anomaly and true
anomaly.
Equation of Time
(due to tilt)

Equation of time due to the Earth’s tilt arises from
the difference between the angle of the mean sun
and the angle of the true sun
Equation of Time
(total)
Total E.O.T. = E.O.T. due to eccentricity
+ E.O.T. due to tilt
Equation of Time
Equation of Time (Earth Analemma)
20
15
Hour Angle (minutes)
10
5
E.O.T. due to Eccentricity
0
E.O.T. due to Tilt
0
100
200
-5
-10
-15
-20
Time (days)
300
400
Total Equation of Time
Solar Declination



Declination of the sun is the angle the sun makes
with the plane of the celestial equator
Varies between +23.45° on June 21st and -23.45°
on December 21st.
On March 21st and September 21st the declination
is zero degrees.
  T (t )  T (equinox)
  arcsin  sin  sin i 
δ = declination
T = true anomaly
i = obliquity (tilt)
Solar Declination
Solar Declination
30
Solar Declination (degrees)
20
March 21: day 80
10
June 21: day 171
0
0
50
100
150
200
250
300
350
400
-10
Sept 21: day 263
Dec 21: day 354
-20
-30
Time (days)
Declination Data
Earth Analemma
Earth Analemma Data
30
Solar Declination (degrees)
20
10
0
-20
-15
-10
-5
0
5
10
15
20
-10
-20
-30
Hour Angle (minutes)
Analemma Data
Comparisons
Equation of Time (Earth Analemma)
20
15
Hour Angle (minutes)
10
5
0
0
50
100
150
200
250
300
350
400
-5
-10
-15
-20
Time (days)
E.O.T. due to Eccentricity
E.O.T. due to Tilt
Code
Total Equation of Time
www.analemma.com
Comparisons
Earth Analemma Data
30
Solar Declination (degrees)
20
10
0
-20
-15
-10
-5
0
5
10
15
20
-10
-20
-30
Hour Angle (minutes)
Analemma Data
Code
www.analemma.com
References
1.
2.
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4.
Duffett-Smith, Peter. Practical Astronomy with your Calculator.
Cambridge University Press, New York, 1981
Press, William H. et al. Numerical Recipes in Fortran: The art of
Scientific Computing. Cambridge University Press, New York,
1992
Urshel, Robert. “The Analemma: A Graphic Explanation” 11 July
05. Accessed 6 Nov 07. <www.analemma.com>.
Waugh, Albert E. Sundials: Their Theory and Construction. Dover
Publications, Inc., New York, 1973